These are various documents and thesis that I have written during my times in mathematics and physics.
My PhD thesis: Symplectic Geometry of Moduli Spaces of Hurwitz Covers
My masters thesis: Mirzakhani’s recursion relations and the Witten conjecture
My bachelor thesis: From Maxwell’s equations to Yang–Mills
My extended high school thesis (German): Die Hohmann-Ellipse und die Lagrange-Punkte für die Raumfahrt
Hurwitz numbers are weigthed counts of equivalence classes of branched covers of Riemann surfaces with fixed branching profile. Calculating these numbers has been a research problem for over 130 years and people continue to be interested in them. For example, this is due to their connection to various intersection numbers in Deligne--Mumford spaces. I am mostly interested in the symplectic structure of compactified moduli spaces of such Hurwitz covers. These spaces come with an evaluation map to the Deligne--Mumford space of the target surface and this map has some kind of degree given by the Hurwitz numbers. Most of my research revolves around making these arguments well-defined and combining this with Maryam Mirzakhanis work on calcuating Psi-class intersections on Deligne--Mumford space via symplectic geometry to obtain new formulas for Hurwitz numbers.
Although this is not my PhD thesis I am very much interested in applying the theory of dynamical systems and symplectic geometry to the real world, in particular space mission design. Starting with the rescue of the Hiten probe it is well-known that one can use low-energy transfer orbits to move in the solar system. Advantages include fuel savings as well as wider launch windows. In order to find these orbits one needs quite a lot of numerical work as well as understanding of the chaotic behaviour of many-body problems. I am trying on understanding, implementing and perhaps improving some of these methods.